From the course of physics in the 8th grade, you know that the sum of potential (mgh) and kinetic (mv 2/2) energy of a body or system of bodies is called total mechanical (or mechanical) energy.
You also know the conservation law mechanical energy:
- the mechanical energy of a closed system of bodies remains constant if only gravitational and elastic forces act between the bodies of the system and there are no friction forces
The potential and kinetic energy of the system can change, transforming into each other. With a decrease in energy of one type, the energy of another type increases by the same amount, due to which their sum remains unchanged.
Let us confirm the validity of the law of conservation of energy by a theoretical conclusion. To do this, consider the following example. A small steel ball of mass m freely falls to the ground from a certain height. At height h 1 (Fig. 51), the ball has a velocity v 1, and when it decreases to a height h 2, its velocity increases to the value v 2.
Rice. 51. Free fall of a ball to the ground from a certain height
The work of the force of gravity acting on the ball can be expressed both through a decrease in the potential energy of the ball's gravitational interaction with the Earth (E p), and through an increase in the kinetic energy of the ball (E k):
Since the left-hand sides of the equations are equal, their right-hand sides are also equal:
It follows from this equation that when the ball moves, its potential and kinetic energy changed. In this case, the kinetic energy has increased by the same amount as the potential has decreased.
After rearranging the terms in the last equation, we get:
The equation written in this form indicates that the total mechanical energy of the ball during its motion remains constant.
It can be written like this:
E p1 + E k1 = E p2 + E k2. (2)
Equations (1) and (2) represent a mathematical record of the law of conservation of mechanical energy.
Thus, we have theoretically proved that the total mechanical energy of a body (more precisely, a closed system of bodies ball - the Earth) is conserved, that is, it does not change over time.
Consider the application of the law of conservation of mechanical energy for solving problems.
Example 1... An apple weighing 200 g falls from a tree from a height of 3 m. What kinetic energy will it have at a height of 1 m from the ground?
Example 2... The ball is thrown down from a height of h 1 = 1.8 m at a speed of v 1 = 8 m / s. How high h 2 will the ball bounce after hitting the ground? (Disregard energy losses from the movement of the ball and its impact on the ground.)
Questions
- What is called mechanical (total mechanical) energy?
- Formulate the law of conservation of mechanical energy. Write it down as equations.
- Can the potential or kinetic energy of a closed system change over time?
Exercise # 22
- Solve the problem considered in the paragraph from Example 2 without using the law of conservation of mechanical energy.
- An icicle detached from the roof falls from a height of h = 36 m from the ground. What speed v will it have at a height of h = 31 m? (Take g = 10 m / s 2.)
- The ball flies out of the children's spring pistol vertically upward with an initial velocity v 0 = 5 m / s. To what height from the place of departure will it rise? (Take g = 10 m / s 2.)
Exercise
Think of and conduct a simple experiment that clearly demonstrates that a body moves curvilinearly if the speed of movement of this body and the force acting on it are directed along intersecting straight lines. Describe the equipment used, your actions, and the observed results.
Chapter Summary
The most important thing
Below are the names of physical laws and their formulations. The sequence of presentation of the formulations of laws does not correspond to the sequence of their names.
Transfer the names of physical laws to the notebook and in square brackets write the ordinal number of the wording corresponding to the named law.
- Newton's first law (law of inertia);
- Newton's second law;
- Newton's third law;
- the law of universal gravitation;
- momentum conservation law;
- the law of conservation of mechanical energy.
- The acceleration of a body is directly proportional to the resultant forces applied to the body, and inversely proportional to its mass.
- The mechanical energy of a closed system of bodies remains constant if only gravitational and elastic forces act between the bodies of the system and there are no friction forces.
- Any two bodies are attracted to each other with a force directly proportional to the mass of each of them and inversely proportional to the square of the distance between them.
- The vector sum of the impulses of the bodies that make up a closed system does not change over time for any movements and interactions of these bodies.
- There are such frames of reference, relative to which the bodies keep their speed unchanged, if other bodies do not act on them or the actions of other bodies are compensated.
- The forces with which two bodies act on each other are equal in magnitude and opposite in direction.
test yourself
Complete the tasks offered in the electronic application.
The total mechanical energy of a closed system of bodies remains unchanged
The energy conservation law can be represented as
If the forces of friction act between the bodies, then the law of conservation of energy changes. The change in the total mechanical energy is equal to the work of the friction forces
Consider the free fall of a body from a certain height h1... The body is not moving yet (let's say we are holding it), the speed is zero, the kinetic energy is zero. The potential energy is maximum, since now the body is higher than everything from the earth than in state 2 or 3.
In state 2, the body has kinetic energy (since it has already developed speed), but at the same time potential energy decreased, since h2 is less than h1. Part of the potential energy has passed into kinetic energy.
State 3 is the state just before stopping. The body, as it were, just touched the ground, while the speed is maximum. The body has maximum kinetic energy. The potential energy is zero (the body is on the ground).
The total mechanical energies are equal, if we neglect the force of air resistance. For example, the maximum potential energy in state 1 is equal to the maximum kinetic energy in state 3.
And where does the kinetic energy then disappear? Disappears without a trace? Experience shows that mechanical movement never disappears without a trace and never arises by itself. During deceleration of the body, the surfaces were heated. As a result of the action of friction forces, the kinetic energy did not disappear, but turned into the internal energy of the thermal motion of molecules.
In any physical interactions, energy does not arise and does not disappear, but only transforms from one form to another.
The main thing to remember
1) The essence of the law of conservation of energy
The general form of the law of conservation and transformation of energy is
Studying thermal processes, we will consider the formula 
In the study of thermal processes, the change in mechanical energy is not considered, that is
The uniformity of time (shear symmetry) leads to conservation law energy : in any process total energy the isolated system does not change; energy can only be converted from one type to another and transferred from one body of the system to another. The law of conservation of energy is a fundamental law of nature that is fulfilled at all structural levels of the organization of matter. There are no phenomena and processes for which this law would not take place. Violation of the law of conservation of energy would indicate a violation of the homogeneity of time.
All phenomena and processes in nature - from the simplest to the most complex - proceed with the conservation of energy. The total supply of energy in the Universe from the moment of its formation to the present day remains constant. The emergence of highly ordered structures (from atoms and molecules to stars and galaxies) and the phenomenon of life are associated with the successive transformations of some forms of energy into others. Part of the energy necessarily goes into the lowest form - heat.
A particular case is of great importance for the practical activity of a person - mechanical energy conservation law carried out in the field of conservative forces.
Conservative is called a force whose work does not depend on the trajectory, but is determined by the initial and final states of the system. The work of a conservative force along a closed path is zero. Conservative are the force of gravity, elasticity, the force of interaction of electric charges, etc. The force, the work of which depends on the trajectory of movement of the body from one point to another, is called dissipative. An example of a dissipative force is frictional force; the work of the friction force along any closed trajectory is less than zero. Force fields in which conservative forces act (for example, a gravitational field or an elastic field) are called potential.
Mechanical energy conservation law: in a system of bodies between which only conservative forces act, the total mechanical energy is conserved (does not change over time)
E m = T+P= const . (2.3.15)
In conservative systems, transformations of kinetic energy into potential and vice versa occur, while the total mechanical energy remains constant.
In dissipative systems, mechanical energy gradually decreases due to transformation into other (non-mechanical) forms. This process is called dissipation (or dissipation) of energy. So, if there is a frictional force in a mechanical system, then mechanical energy is partially converted into heat.
Control questions
1 What is symmetry? Give examples of symmetry operations.
2 Formulate Noether's theorem. What is the relationship between symmetry and conservation laws?
3 Formulate the law of conservation of momentum. What property of space is this law associated with?
4 Give examples of phenomena that can be explained by the law of conservation of momentum.
5 Formulate the law of conservation of angular momentum. What property of space is this law associated with?
6 Give examples of the phenomena explained by the law of conservation of angular momentum.
This video tutorial is intended for independent acquaintance with the topic "The Law of Conservation of Mechanical Energy". First, we give the definition of total energy and a closed system. Then we will formulate the Law of Conservation of Mechanical Energy and consider in which areas of physics it can be applied. We will also give a definition of work and learn how to define it by considering the formulas associated with it.
The topic of the lesson is one of the fundamental laws of nature - mechanical energy conservation law.
We talked earlier about potential and kinetic energy, and also about the fact that a body can have both potential and kinetic energy together. Before talking about the law of conservation of mechanical energy, let's remember what total energy is. Full mechanical energy called the sum of the potential and kinetic energies of the body.
Let's also remember what is called a closed system. Closed system- this is such a system in which there is a strictly defined number of interacting bodies and no other bodies from the outside act on this system.
When we have decided on the concept of total energy and a closed system, we can talk about the law of conservation of mechanical energy. So, the total mechanical energy in a closed system of bodies interacting with each other by means of gravitational forces or elastic forces (conservative forces) remains unchanged for any movement of these bodies.
We have already studied the Law of Conservation of Momentum (MMP):
It often happens that the set tasks can be solved only with the help of the laws of conservation of energy and momentum.
It is convenient to consider the conservation of energy using the example of the free fall of a body from a certain height. If a body is at rest at a certain height relative to the earth, then this body has potential energy. As soon as the body begins to move, the height of the body decreases, and the potential energy decreases. At the same time, the speed begins to increase, kinetic energy appears. When the body approached the ground, the height of the body is equal to 0, the potential energy is also equal to 0, and the maximum kinetic energy of the body will be. This is where the transformation of potential energy into kinetic energy is seen (Fig. 1). The same can be said about the movement of the body on the contrary, from bottom to top, when the body is thrown vertically upward.
Rice. 1. Free fall of the body from a certain height
Additional task 1. "On the fall of a body from a certain height"
Problem 1
Condition
The body is at a height from the surface of the Earth and begins to fall freely. Determine the speed of the body at the moment of contact with the ground.
Solution 1:
Initial body speed. Need to find .
Consider the law of conservation of energy.
Rice. 2. Body movement (task 1)
At the top, the body has only potential energy: . When the body approaches the ground, the body's height above the ground will be equal to 0, which means that the body's potential energy has disappeared, it has turned into kinetic energy:
According to the law of conservation of energy, we can write:
Body weight is reduced. Transforming the above equation, we get:.
The final answer would be:. If we substitute all the value, we get: .
Answer: .
An example of a solution to the problem:
Rice. 3. An example of registration of the solution to problem No. 1
This problem can be solved in one more way, like vertical movement with the acceleration of gravity.
Solution 2 :
Let us write the equation of motion of the body in projection onto the axis:
When the body approaches the surface of the Earth, its coordinate will be equal to 0:
Acceleration due to gravity is preceded by a "-" sign because it is directed against the selected axis.
Substituting the known values, we find that the body fell over time. Now let's write the equation for the speed:
Assuming the gravitational acceleration to be equal, we get:
The minus sign means that the body is moving against the direction of the selected axis.
Answer: .
An example of formalizing the solution to problem No. 1 in the second way.
Rice. 4. An example of registration of the solution to problem number 1 (method 2)
Also, to solve this problem, you could use a formula that does not depend on time:
Of course, it should be noted that we considered this example taking into account the absence of friction forces, which in reality act in any system. Let's turn to the formulas and see how the law of conservation of mechanical energy is written:
Additional task 2
The body falls freely from a height. Determine at what height the kinetic energy is equal to one third of the potential ().
Rice. 5. Illustration for problem no. 2
Solution:
When a body is at a height, it has potential energy, and only potential. This energy is determined by the formula: 
.
This will be the total energy of the body.
When the body begins to move downward, potential energy decreases, but at the same time kinetic energy increases. At the height to be determined, the body will already have a certain velocity V. For the point corresponding to the height h, the kinetic energy has the form:
The potential energy at this altitude will be indicated as follows: 
.
According to the law of conservation of energy, we conserve total energy. This energy 
remains constant. For a point, we can write the following ratio: (according to Z.S.E.).
Recalling that the kinetic energy is, according to the problem statement, we can write the following:.
Please note: the mass and the acceleration of gravity are reduced, after simple transformations we get that the height at which this ratio is satisfied is.
Answer:
An example of task design 2.
Rice. 6. Registration of the solution to problem No. 2
Imagine that a body in a certain frame of reference has kinetic and potential energy. If the system is closed, then with any change a redistribution occurred, a transformation of one type of energy into another, but the total energy remains the same in value (Fig. 7).
Rice. 7. The law of conservation of energy
Imagine a situation where a car is moving on a horizontal road. The driver turns off the engine and continues driving with the engine turned off. What happens in this case (fig. 8)?
Rice. 8. Vehicle movement
In this case, the car has kinetic energy. But you know perfectly well that over time the car will stop. Where did the energy go in this case? After all, the potential energy of the body in this case also did not change, it was some kind of constant value relative to the Earth. How did the energy change come about? In this case, the energy was used to overcome the frictional forces. If friction occurs in a system, then it also affects the energy of this system. Let's see how the energy change is recorded in this case.
Energy changes, and this energy change is determined by work against frictional force. We can determine the work of the friction force using the formula, which is known from the 7th class (force and displacement are opposite):
So, when we talk about energy and work, we must understand that every time we must take into account the fact that part of the energy is spent on overcoming friction forces. Work is being done to overcome the frictional forces. Work is a quantity that characterizes the change in the energy of the body.
In conclusion of the lesson, I would like to say that work and energy are inherently related quantities through acting forces.
Additional Objective 3
Two bodies - a block of mass and a plasticine ball of mass - move towards each other at the same speeds (). After the collision, the plasticine ball stuck to the bar, the two bodies continue to move together. Determine how much of the mechanical energy has turned into the internal energy of these bodies, taking into account the fact that the mass of the bar is 3 times greater than the mass of the plasticine ball ().
Solution:
The change internal energy can be designated. As you know, there are several types of energy. In addition to mechanical energy, there is also thermal, internal energy.
Imagine a roaring waterfall. Powerful streams of water make a terrible noise, drops sparkle in the sun, foam whitens. Nice, isn't it?
Conversion of one type of mechanical energy into another
And what do you think, does this element rushing downward have energy? No one will argue that yes. But what kind of energy will the water have - kinetic or potential? And here it turns out that neither the first nor the second answer will be correct. And the answer will be correct - falling water has both types of energy. That is, one and the same body can have both types of energy. Their sum is called the total mechanical energy of the body: E = E_k + E_p. Moreover, in this case, water not only possesses both types of energy, but their value changes in the course of water movement. When our water is at the top of the waterfall and has not yet begun to fall, then it has the maximum potential energy. The kinetic energy in this case is equal to zero. When the water begins to fall down, it has kinetic energy of motion. In the course of movement downward, the potential energy decreases, since the height decreases, and the kinetic, on the contrary, increases, since the speed of the water falling increases. That is, there is a transformation of one type of energy into another. In this case, the total mechanical energy is conserved. This is the law of conservation and transformation of energy.
Total mechanical energy conservation law
The total mechanical energy conservation law states: the total mechanical energy of the body, which is not acted upon by the forces of friction and resistance, remains unchanged in the process of its movement. When, for example, sliding friction is present, the body is forced to spend part of the energy to overcome it, and the energy will naturally decrease. Therefore, in reality, when transferring energy, there are almost always losses that have to be taken into account.
The energy conservation law can be represented as a formula. If we denote the initial and final energy of the body as E_1 and E_2, then the law of conservation of energy can be expressed as follows: E_1 = E_2. At the initial moment of time, the body had a velocity v_1 and a height h_1:
E_1 = (mv_1 ^ 2) / 2 + mgh_1.
At the final moment of time with a speed v_2 at an altitude of h_2, the energy
E_2 = (mv_2 ^ 2) / 2 + mgh_2.
In accordance with the law of conservation of energy:
(mv_1 ^ 2) / 2 + mgh_1 = (mv_2 ^ 2) / 2 + mgh_2.
If we know the initial values of speed and energy, then we can calculate the final speed at height h, or, conversely, find the height at which the body will have a given speed. In this case, body weight does not matter, since it will be reduced from the equation.
Energy can also be transferred from one body to another. So, for example, when an arrow is released from a bow, the potential energy of the bowstring turns into kinetic energy flying arrow.
